Problem: All of the 4th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$2.50$ each for students, and the group paid $$36.50$ in total. The next month, the same group visited a science museum where the tickets cost $$32.00$ each for teachers and $$9.50$ each for students, and the group paid $$143.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+2.5y = 36.5}$ ${32x+9.5y = 143.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-32x-10y = -146}$ ${32x+9.5y = 143.5}$ Add the top and bottom equations together. $ -0.5y = -2.5 $ $ y = \dfrac{-2.5}{-0.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {8x+2.5y = 36.5}$ to find $x$ ${8x + 2.5}{(5)}{= 36.5}$ $8x+12.5 = 36.5$ $8x = 24$ $x = \dfrac{24}{8}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {32x+9.5y = 143.5}$ and get the same answer for $x$ ${32x + 9.5}{(5)}{= 143.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.